Short answers, pulled from the story.
Pure Land Buddhism is a broad branch of Mahayana Buddhism focused on achieving rebirth in a Buddha's pure land after death. The most popular destination is Sukhavati, the pure land of Buddha Amitabha, which is understood as an ideal environment for attaining full Buddhahood. The tradition is practiced primarily through mindfulness of the Buddha, especially by reciting Amitabha's name.
Huiyuan founded Donglin Temple at Mount Lu in 402 and is traditionally credited as the first patriarch of Chinese Pure Land Buddhism. The monk Shandao (613-681), however, is identified by scholar Charles B. Jones as the true founder because he was the first to state clearly that ordinary beings can attain rebirth in the Pure Land through Amitabha's vow alone.
East Asian Pure Land relies primarily on three Mahayana scriptures: the Sutra of Amitayus (Longer Sukhavativyuha), the Contemplation Sutra, and the Amitabha Sutra (Shorter Sukhavativyuha). In Chinese Pure Land Buddhism, these three are combined with two sutra chapters and a treatise by Vasubandhu to form a canon of six foundational texts.
Nianfo (Chinese: niànfó) refers to mindfulness of the Buddha, most commonly practiced by reciting the name of Amitabha. The monk Fazhao, who died around 820, standardized the Chinese classic chant of na-mo a-mi-tuo fo. Shandao (613-681) established oral recitation of Amitabha's name as the central Pure Land practice, arguing that even ten sincere repetitions were sufficient for rebirth in Sukhavati.
Pure Land thought entered Korean Buddhism from China during the Unified Silla period (668-935). The most influential figure in this development was Wonhyo (617-686), who promoted nianfo practice widely and wrote ten texts on Pure Land Buddhism. All later Korean writings on Pure Land trace back through Wonhyo's influence.
Self-power refers to spiritual progress achieved through one's own study, meditation, and moral effort. Other-power refers to reliance on the compassionate power of a Buddha, especially Amitabha. The patriarch Tanluan developed this distinction, arguing that in the current degenerate age, the classic bodhisattva path relying on self-power was too difficult, and that faith in Amitabha's other-power offered an accessible path to awakening.